Problem: Determine how many solutions exist for the system of equations. ${-3x+y = 6}$ ${y = 6+3x}$
Explanation: Convert both equations to slope-intercept form: ${-3x+y = 6}$ $-3x{+3x} + y = 6{+3x}$ $y = 6+3x$ ${y = 3x+6}$ ${y = 6+3x}$ ${y = 3x+6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x+6}$ ${y = 3x+6}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-3x+y = 6}$ is also a solution of ${y = 6+3x}$, there are infinitely many solutions.